Patricia Gaitan

Inverse problems for a 2 × 2 reaction-diffusion system using a Carleman estimate with one observation

For a two by two reaction-diffusion system on a bounded domain we give a simultaneous stability result for one coefficient and for the initial conditions. The key ingredient is a global Carleman-type estimate with a single observation acting on a subdomain.

Inverse problem for a parabolic system with two components by measurements of one component

We consider a 2 × 2 system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time θ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-degeneracy condition at θ for the two components and for it, we can approximately control the two components of the 2 × 2 system by inputs to only one component. Such approximate controllability is proved also by our new Carleman estimate. Finally, we establish a Carleman estimate for a 3 × 3 system for parabolic equations with coupling of zeroth-order terms by one component to show the corresponding approximate controllability with a control to one component.

Stability of Discontinuous Diffusion Coefficients and Initial Conditions in an Inverse Problem for the Heat Equation

We consider the heat equation with a discontinuous diffusion coefficient and give uniqueness and stability results for both the diffusion coefficient and the initial condition from a measurement of the solution on an arbitrary part of the boundary and at some arbitrary positive time. The key ingredient is the derivation of a Carleman-type estimate. The diffusion coefficient is assumed to be discontinuous across an interface with a monotonicity condition.

Inverse Problem for the Schrödinger Operator in an Unbounded Strip

Abstract We consider the operator H := i∂t + ∇ . (c∇) in an unbounded strip Ω in ℝ2, where . We prove an adapted global Carleman estimate and an energy estimate for this operator. Using these estimates, we give a stability result for the diffusion coefficient c(x, y).

A stability result for a time-dependent potential in a cylindrical domain

We consider the operator H := ∂t − Δ + V in a 2D or 3D cylindrical domain with mixed Dirichlet and Neumann conditions. With an adapted global Carleman estimate with singular weight functions, we give a stability result for the time-dependent part of the potential for this particular geometry.

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ∂ t u = div (γ∇ x u) in (0, T) × Ω, u = f on (0, T) × ∂Ω, u| t=0 = u 0, in a bounded domain Ω ⊂ ℝ n , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k 2 (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω∖D(t)). Fix a direction e* ∈ 𝕊 n−1 arbitrarily. Assuming that ∂D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ∂ν u(t, x)|(0, T)×∂Ω. The knowledge of the initial data u 0 is not used in the proof. If we know min0≤t≤T (sup x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.

Inverse problem for a free transport equation using Carleman estimates

This paper is devoted to prove a stability result for an absorption coefficient for a free transport equation in a smooth domain . The result is obtained using a global Carleman estimate with only one observation on a part of the boundary.

Identification of two coefficients with data of one component for a nonlinear parabolic system

In this article, we consider a nonlinear parabolic system with two components and prove a stability estimate of Lipschitz type in determining two coefficients of the system by data of only one component. The main idea for the proof is a Carleman estimate.

Simultaneous Identification of the Diffusion Coefficient and the Potential for the Schrodinger Operator with only one Observation

This paper is devoted to proving a stability result for two independent coefficients for a Schrodinger operator in an unbounded strip. The result is obtained with only one observation on an unbounded subset of the boundary and the data of the solution at a fixed time on the whole domain.

Inverse Problems for Time-Dependent Singular Heat Conductivities---One-Dimensional Case

We consider an inverse boundary value problem for the heat equation on the interval